This page is explaining how to characterize the homogeneity of a bulk solids mixture by calculating a coefficient of variation (CV) for a representative property.

1. Calculation of the coefficient of variation CV for homogeneity

How to calculate the coefficient of variation ?

After having completed the mixing and having performed the sampling, the samples taken are analyzed regarding the presence of the tracer. Then results are recorded and statistics can be performed.

The mixture, regarding the tracer, will be defined according to the mean concentration of the tracer and to the standard deviation of the tracer concentration. Mean and standard deviation will be used to calculate the relative standard deviation of the mix, that will be the value assimilated to "homogeneity".

Equation 1 : Relative Standard Deviation of the mixture

S is the samples standard deviation, it is not the actual standard deviation since only an estimation can be given from the samples. μ is the arithmetical average of the samples concentration, thus as well calculated from the samples.

This value is often expressed in % and called also the coefficient of variation.

How to calculate CV % :

Equation 2 : Formula of coefficient of variation of the mixture

The value obtained is then compared to the specification.

WARNING : the CV that will be obtained has actually several components, and some of these components need to be calculated in order to estimate the actual homogeneity variance.

The sample variance is calculated thanks to :

S²=S_mix²+S_analytical²+(S_sampling²)

The variability due to sampling is very difficult to determine, thus, in practice, it will be neglected. However, for this assumption to be true it is critical to sample the mix following the methods explained above, and preferably on the free flowing powder, so that the variance due to sampling is small.

The variability due to analysis can either be known, if experiments have been done before, or can be determined for the particular homogeneity validation by doubling the measurement on the same sample.

S_mix can then be calculated. Then CV_mix (%)

2. Is your mixing ok ? Confidence intervals

How to interpret the coefficient of variation ?

WARNING : Once CV_mix(%) calculated, it is not sufficient to compare it to the specification. Indeed, the variance calculated is NOT a true variance but is estimated on the sampling. This estimation must be taken into account by calculating a confidence interval, generally at 95%, which corresponds to 2 sigma on each side of the mean.

The specification can then be compared to the confidence interval :

• If the CV_spec > upper border of the confidence interval : the mixing is successful, the mix is significantly better than the specification
• If CV_spec is in the confidence interval : the mixing may be successful, but it is also possible that the actual CVmix is > CVspec
• If CV_spec is < lower border of the confidence interval : the mixing homogeneity achieved is not good enough for the application.

Figure 1 : Interpretation of coefficient of variation

For the 2 last cases, case by case discussion will be necessary within the factory team :

• Accept the mix
• Reject the mix and look for root causes (mixer speed, filling rate of mixer...)